Relaxtion methods applied to engineering problems II. Basic theory, with application to surveying and to electrical networks, and an extension to gyrostatic systems.

Abstract

1. The “Method of Systematic Relaxation of Constraints” was devised for the determination of stresses in frameworks,—that is in elastic structures having the characteristic that a strained configuration can be specified by attaching values to a finite number of co-ordinates. Recently it has been extended to continuous systems (e. g. beams) on the understanding that a finite number of co-ordinates will define a configuration for practical purposes, though not from a mathematical standpoint. So far the power of the method has been exhibited only in relation to elastic problems: in these its results appear to converge rapidly, judged by a few examples of which the exact solutions were known. The aim of the present paper is fourfold, viz. (1) to prove that relaxation methods, applied systematically to problems of elastic equilibrium, give solutions which converge steadily towards exact results; and hence, by analogy, (2) to show that they are applicable to any “minimal” problem, e. g. the adjustment of errors according to the method of least squares; (3) to notice, as particular examples, the adjustment of errors in level or triangulation surveys, and the partition of electric current in non-inductive networks of conductors; and (4) to discuss the more difficult problem of an inductive network carrying alternating current. This serves as a simple illustration of systems which are governed by equations containing “gyrostatic” or “non-energetic” terms, and which for that reason do not present minimal problems of the usual kind.